A049033 - OEIS

%I #13 Jan 20 2025 06:40:11

%S 1,56,4368,489280,75851776,15658639360,4165906530304,1390437378293760,

%T 569462999991975936,280969831084430721024,164441704270786486861824,

%U 112668650067303149573505024

%N Central factorial numbers: unsigned 2nd subdiagonal of A182867.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

%F E.g.f.: (arcsin x)^6; that is, a_k is the coefficient of x^(2*k+6) in (arcsin x)^6 multiplied by (2*k+6)! and divided by 6!. - Joe Keane (jgk(AT)jgk.org)

%F (-1)^(n-2)*a(n-2) is the coefficient of x^5 in prod(k=0, 2*n, x+2*k-2*n). - _Benoit Cloitre_ and _Michael Somos_, Nov 22 2002

%e (arcsin x)^6 = x^6 + x^8 + 13/15*x^10 + 139/189*x^12 + ...

%p A049033 := proc(n)

%p     arcsin(x)^6;

%p     coeftayl(%,x=0,2*n+6)*(2*n+6)!/6! ;

%p end proc:

%p seq(A049033(n),n=0..20) ; # _R. J. Mathar_, Jan 20 2025

%Y Cf. A001824, A001825, A002455.

%Y Equals 4^n * A001820(n).

%K nonn

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)